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Register a new userExplosive transitions in complex networks structure and dynamics: percolation and synchronization
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Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in com- plex networks structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the networks structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and de- bates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.
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This Letter investigates the nature of synchronization in multilayered and multiplexed populations in which the interlayer interactions are randomly pinned. First, we show that a multilayer network constructed by setting up all-to-all interlayer connections between the two populations leads to explosive synchronization in the two populations successively, leading to the coexistence of coherent and incoherent populations forming chimera states. Second, a multiplex formation of the two populations in which only the mirror nodes are interconnected espouses explosive transitions in the two populations concurrently. The emergence of explosive synchronization is substantiated with rigorous mean-field calculations demonstrating the existence of a bistable region. The random pinning in the interlayer interactions concerns the practical problems where the impact of dynamics of one network on that of other interconnected networks remains elusive, as is the case for many real-world systems.
The phenomenon of explosive synchronization, which originates from hypersensitivity to small perturbation caused by some form of frustration prevailed in various physical and biological systems, has been shown to lead events of cascading failure of the power grid to chronic pain or epileptic seizure in the brain. Furthermore, networks provide a powerful model to understand and predict the properties of a diverse range of real-world complex systems. Recently, a multilayer network has been realized as a better suited framework for the representation of complex systems having multiple types of interactions among the same set of constituents. This article shows that by tuning the properties of one layer (network) of a multilayer network, one can regulate the dynamical behavior of another layer (network). By taking an example of a multiplex network comprising two different types of networked Kuramoto oscillators representing two different layers, this article attempts to provide a glimpse of opportunities and emerging phenomena multiplexing can induce which is otherwise not possible for a network in isolation. Here we consider explosive synchronization to demonstrate the potential of multilayer networks framework. To the end, we discuss several possible extensions of the model considered here by incorporating real-world properties.
It is known that intra-layer adaptive coupling among connected oscillators instigates explosive synchronization (ES) in multilayer networks. Taking an altogether different cue in the present work, we consider inter-layer adaptive coupling in a multiplex network of phase oscillators and show that the scheme gives rise to ES with an associated hysteresis irrespective of the network architecture of individual layers. The hysteresis is shaped by the inter-layer coupling strength and the frequency mismatch between the mirror nodes. We provide rigorous mean-field analytical treatment for the measure of global coherence and manifest they are in a good match with respective numerical assessments. Moreover, the analytical predictions provide a complete insight into how adaptive multiplexing suppresses the formation of a giant cluster, eventually giving birth to ES. The study will help in spotlighting the role of multiplexing in the emergence of ES in real-world systems represented by multilayer architecture. Particularly, it is relevant to those systems which have limitations towards change in intra-layer coupling strength.
Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by excessive synchronization. This indicates that coherence alone can not be enough to explain all the structural features observed in many real-world networks. In this work, we propose an adaptive network model where the dynamical evolution of the node states towards synchronization is coupled with an evolution of the link weights based on an anti-Hebbian adaptive rule, which accounts for the presence of inhibitory effects in the system. We found that the emergent networks spontaneously develop the structural conditions to sustain explosive synchronization. Our results can enlighten the shaping mechanisms at the heart of the structural and dynamical organization of some relevant biological systems, namely brain networks, for which the emergence of explosive synchronization has been observed.
Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper, we show how in a network of identical dynamical systems, nodes belonging to the same degree class differentiate in the same manner visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics including toroidal chaos and how it depends on the coupling function. This study provides new insights to understand better strategies for network identification and control or to devise effective methods for network inference.
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